Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD Marc Klegrewe and Wolfgang Unger Bielefeld University Lattice 2018 East Lansing, 27th July 2018 Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 1 / 14
QCD in the Strong Coupling Limit QCD in the Strong Coupling Limit QCD in the Strong Coupling Limit QCD in the Strong Coupling Limit QCD in the Strong Coupling Limit Study regime where sign problems can be made mild: [Wolff & Rossi, 1984] ⇒ limit of infinite gauge coupling Problem β = 2 N c g → ∞ , g 2 → 0 Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 2 / 14
QCD in the Strong Coupling Limit QCD in the Strong Coupling Limit QCD in the Strong Coupling Limit QCD in the Strong Coupling Limit QCD in the Strong Coupling Limit Study regime where sign problems can be made mild: [Wolff & Rossi, 1984] ⇒ limit of infinite gauge coupling Problem β = 2 N c g → ∞ , g 2 → 0 A change of integration order results in SC-partition function for staggered fermions: � � N c ! � ( N c − k b )! � n x ! (2 am q ) n x γ 2 k b δ µ 0 Z SC = w ( l , µ ) N c ! k b ! { n , k , l } x b =( x ,µ ) l � �� � � �� � � �� � monomers mesonic hoppings/dimers baryonic hoppings Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 2 / 14
QCD in the Strong Coupling Limit QCD in the Strong Coupling Limit QCD in the Strong Coupling Limit QCD in the Strong Coupling Limit QCD in the Strong Coupling Limit Study regime where sign problems can be made mild: [Wolff & Rossi, 1984] ⇒ limit of infinite gauge coupling Problem β = 2 N c g → ∞ , g 2 → 0 A change of integration order results in SC-partition function for staggered fermions: � � N c ! � ( N c − k b )! � n x ! (2 am q ) n x γ 2 k b δ µ 0 Z SC = w ( l , µ ) N c ! k b ! { n , k , l } x b =( x ,µ ) l � �� � � �� � � �� � monomers mesonic hoppings/dimers baryonic hoppings Fully combinatorial problem, restricted by Grassmann constraint : � � n x + k x µ = N c , l x µ = 0 , ∀ x ± µ ± µ In the following restrict to chiral limit where monomer density � n � = 0 Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 2 / 14
Continuous Time limit within Strong Coupling QCD Continuous Time limit within Strong Coupling QCD Continuous Time limit within Strong Coupling QCD Continuous Time limit within Strong Coupling QCD Continuous Time limit within Strong Coupling QCD First: Introduction of anisotropy for continuous temperature variation: aT = 1 N τ ⇒ aT = ξ ( γ ) N τ , ξ ( γ ) = a / a τ � �� � anisotropy parameter Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 3 / 14
Continuous Time limit within Strong Coupling QCD Continuous Time limit within Strong Coupling QCD Continuous Time limit within Strong Coupling QCD Continuous Time limit within Strong Coupling QCD Continuous Time limit within Strong Coupling QCD First: Introduction of anisotropy for continuous temperature variation: aT = 1 N τ ⇒ aT = ξ ( γ ) N τ , ξ ( γ ) = a / a τ � �� � anisotropy parameter Second: Gamma dependence of ξ ( γ ) non trivial: [de Forcrand, Unger & Vairinhos, 2018] γ 2 ξ ( γ ) ≈ κγ 2 + , κ = 0 . 781 for SU(3) 1 + λγ 4 Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 3 / 14
Continuous Time limit within Strong Coupling QCD Continuous Time limit within Strong Coupling QCD Continuous Time limit within Strong Coupling QCD Continuous Time limit within Strong Coupling QCD Continuous Time limit within Strong Coupling QCD First: Introduction of anisotropy for continuous temperature variation: aT = 1 N τ ⇒ aT = ξ ( γ ) N τ , ξ ( γ ) = a / a τ � �� � anisotropy parameter Second: Gamma dependence of ξ ( γ ) non trivial: [de Forcrand, Unger & Vairinhos, 2018] γ 2 ξ ( γ ) ≈ κγ 2 + , κ = 0 . 781 for SU(3) 1 + λγ 4 Definition of the Continuous Time Limit as: = κγ 2 N τ → ∞ , γ → ∞ with ξ ( γ ) N τ = aT fixed N τ Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 3 / 14
Continuous Time limit within Strong Coupling QCD Continuous Time limit within Strong Coupling QCD Continuous Time limit within Strong Coupling QCD Continuous Time limit within Strong Coupling QCD Continuous Time limit within Strong Coupling QCD First: Introduction of anisotropy for continuous temperature variation: aT = 1 N τ ⇒ aT = ξ ( γ ) N τ , ξ ( γ ) = a / a τ � �� � anisotropy parameter Second: Gamma dependence of ξ ( γ ) non trivial: [de Forcrand, Unger & Vairinhos, 2018] γ 2 ξ ( γ ) ≈ κγ 2 + , κ = 0 . 781 for SU(3) 1 + λγ 4 Definition of the Continuous Time Limit as: = κγ 2 N τ → ∞ , γ → ∞ with ξ ( γ ) N τ = aT fixed N τ Continuous Time partition function: � 1 � � � e µ B B / T ˆ ν N T Z CT ( T ) = T 2 aT G ′ ∈ Γ k k ∈ 2 N � � with k = k b , N T = n T ( x ) b =( x , ˆ x i ) Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 3 / 14
Benefits and Comments on Continuous Time Limit Benefits and Comments on Continuous Time Limit Benefits and Comments on Continuous Time Limit Benefits and Comments on Continuous Time Limit Benefits and Comments on Continuous Time Limit • No discretization errors due to finite N τ • Only one parameter left (temperature T) • Baryons become static for N c ≥ 3 ⇒ no extend in spatial direction ⇒ Sign problem is absent • Baryons are massive (even though in chiral limit) • No multiple spatial dimers (suppressed by γ ) • Faster algorithm for medium to large temporal extends Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 4 / 14
Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm • Worm-type Monte Carlo Algorithm [Adams & Chandrasekharan, 2003] • absorption (even) and emission (odd) site decomposition of lattice Mesonic worm update: • Place tail of mesonic worm on lattice at absorption site ⇒ Violation of Grassmann constraint ⇒ propagate head restoration of Grassmann constraint if head at emission site t Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 5 / 14
Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm • Worm-type Monte Carlo Algorithm [Adams & Chandrasekharan, 2003] • absorption (even) and emission (odd) site decomposition of lattice Mesonic worm update: • Place tail of mesonic worm on lattice at absorption site ⇒ Violation of Grassmann constraint ⇒ propagate head restoration of Grassmann constraint if head at emission site H T t Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 5 / 14
Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm • Worm-type Monte Carlo Algorithm [Adams & Chandrasekharan, 2003] • absorption (even) and emission (odd) site decomposition of lattice Mesonic worm update: • Place tail of mesonic worm on lattice at absorption site ⇒ Violation of Grassmann constraint ⇒ propagate head restoration of Grassmann constraint if head at emission site H T t Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 5 / 14
Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm • Worm-type Monte Carlo Algorithm [Adams & Chandrasekharan, 2003] • absorption (even) and emission (odd) site decomposition of lattice Mesonic worm update: • Place tail of mesonic worm on lattice at absorption site ⇒ Violation of Grassmann constraint ⇒ propagate head restoration of Grassmann constraint if head at emission site H t T t Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 5 / 14
Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm • Worm-type Monte Carlo Algorithm [Adams & Chandrasekharan, 2003] • absorption (even) and emission (odd) site decomposition of lattice Mesonic worm update: • Place tail of mesonic worm on lattice at absorption site ⇒ Violation of Grassmann constraint ⇒ propagate head restoration of Grassmann constraint if head at emission site e T H ? a t Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 5 / 14
Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm • Worm-type Monte Carlo Algorithm [Adams & Chandrasekharan, 2003] • absorption (even) and emission (odd) site decomposition of lattice Mesonic worm update: • Place tail of mesonic worm on lattice at absorption site ⇒ Violation of Grassmann constraint ⇒ propagate head restoration of Grassmann constraint if head at emission site H T a t Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 5 / 14
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